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Kaluza-Klein theories
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1987 Rep. Prog. Phys. 50 1087
(http://iopscience.iop.org/0034-4885/50/9/001)
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Rep. Prog. Phys. 50 (1987) 1087-1170. Printed in the UK 
Kaluza-Klein theories 
David Bailint and Alex Love$ 
t School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK 
$ Department of Physics, R H B New College, University of London, Egham, UK 
Abstract 
Kaluza-Klein theory is developed starting from the simplest example in which a single 
extra spatial dimension is compactified to a circle, and a single Abelian gauge field 
emerges in four dimensions from the higher-dimensional metric. This is generalised 
to greater dimensionality whence non-Abelian gauge groups may be obtained, and 
possible mechanisms for achieving the compactification of the extra spatial dimensions 
are discussed. The spectrum of particles appearing in four dimensions is discussed 
with particular emphasis on the spectrum of light fermions, and the constraints arising 
from cancellation of anomalies when explicit higher-dimensional gauge fields are 
present are studied. Cosmological aspects of these theories are described, including 
possible mechanisms for cosmological inflation, and relic heavy particles. Finally, an 
introductory account of Kaluza-Klein supergravity is given leading towards superstring 
theory. 
This review was received in July 1986. 
0034-4885/87/091087 + 84$09.00 0 1987 IOP Publishing Ltd 1087 
1088 D Bailin and A Love 
Contents 
1. Five-dimensional Kaluza-Klein theory 
1.1. Introduction 
1.2. The five-dimensional theory 
1.3. Abelian gauge transformations 
1.4. Effective four-dimensional action 
1.5. Mass eigenstates 
1.6. Charge quantisation 
1.7. Scalar field from mstric 
2. (4-t D)-dimensional Kaluza-Klein theories 
2.1. Isometry group of a manifold 
2.2. Non-Abelian gauge transformations 
2.3. Effective four-dimensional action 
2.4. Graviton-scalar sector 
2.5. Differential geometry 
2.6. The manifolds MPqr 
3. Compactification mechanisms 
3.1. General considerations 
3.2. Freund-Rubin compactification 
3.3. 
3.4. 
3.5. Monopole solutions 
3.6. Instanton solutions 
3.7. 
3.8. Non-minimal gravitational actions 
3.9. 
3.10. Stability of compactification for Einstein-Yang-Mills theories 
4.1. Geometric interpretation 
4.2. Absolute values of gauge coupling constants 
4.3. Ratios of gauge coupling constants 
5. Particle spectrum of Kaluza-Klein theories 
5.1. Boson spectrum 
5.2. Kac-Moody symmetries 
5.3. Fermions (Witten 1983) 
5.4. Anomalies 
6.1. Introduction 
6.2. Late-time solutions (Bailin er al 1984) 
6.3. Early-time solutions 
6.4. Survival of massive modes (Kolb and Slansky 1984) 
6.5. Kaluza-Klein monopoles 
Quantum fluctuations in massless higher-dimensional fields 
Compactification due to explicit gauge fields 
Generalised monopole and instanton solutions 
Stability of compactification of pure Kaluza-Klein theories 
4. Gauge coupling constants 
6. Kaluza-Klein cosmology 
Page 
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Kaluza-Klein theories 
7 . Kaluza- Klein supergravity 
7.1. Eleven-dimensional supergravity 
7.2. Compactification of eleven-dimensional supergravity 
7.3. Unbroken supersymmetries in four dimensions 
7.4. Minimal ten-dimensional supergravity 
7.5. Anomaly-free ten-dimensional supergravity 
8. Conclusions 
References 
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Kaluza- Klein theories 1091 
1. Five-dimensional Kaluza-Klein theory 
1 . 1 . Introduction 
A theory unifying gravitation and electromagnetism using five-dimensional Riemannian 
geometry (Kaluza 1921, Klein 1926) and its higher-dimensional generalisations to 
include weak and strong interactions (DeWitt 1964, Kerner 1968, Trautman 1970, Cho 
1975, Cho and Freund 1975) have become a focus of attention for many particle 
physicists in the past few years. This revival of interest in Kaluza-Klein theory stemmed 
in the first instance from work in string theories (Scherk and Schwarz 1975, Cremmer 
and Scherk 1977), and then from the usefulness of extra spatial dimensions in the 
construction of N = 8 supergravity theory (Cremmer et a1 1978, Cremmer and Julia 
1978). In these contexts, it would have been possible to regard the extra spatial 
dimensions as a mathematical device. However, the above authors took the potentially 
more fruitful approach of considering them to be genuine physical dimensions which 
we do not normally observe because they have compactified down to a very small scale 
(spontaneous compactification). Extra temporal dimensions are undesirable for several 
reasons. Firstly, there would be tachyons observed in four dimensions. (See the 
analysis given in § 1.5.) Secondly, there would be closed timelike loops leading to 
world lines which violate causality. Thirdly, the sign of the Maxwell action would be 
incorrect. (See the analysis in § 1.4.) 
Though the renaissance of Kaluza-Klein theory has received a considerable impetus 
from the possible relevance to supergravity, many theorists have taken the view that 
extra spatial dimensions may be an ingredient in the unification of all interactions 
even if supergravity should eventually have to be relinquished. In this review, we shall 
avoid, as far as possible, reliance on supergravity as a framework for Kaluza-Klein 
theory. Instead we shall try to emphasise those aspects of such theories which might 
be of importance for unification of interactions with or without supergravity. Our 
approach is pedagogical and directed primarily towards readers without an extensive 
background in supergravity theories. Accordingly, we have been selective in the 
material included, and consequently in that section of the literature to which we refer. 
The reader may restore the imbalance against work deeply rooted in supergravity 
theory by a reading of one of the recent excellent review papers which have approached 
the subject from that standpoint (Duff et a1 1986, Englert and Nicolai 1983). 
1.2. The Jive-dimensional theory 
Five-dimensional Kaluza-Klein theory (Kaluza 1921, Klein 1926) unifies electromag- 
netism with gravitation by starting from a theory of Einstein gravity in five dimensions. 
Thus, the initial theory has five-dimensional general coordinate invariance. However, 
it is assumed that one of the spatial dimensions compactifies so as to have the geometry 
of a circle S' of very small radius. Then, there is a residual four-dimensional general 
coordinate invariance, and, as we see in 8 1.3, an Abelian gauge invariance associated 
with transformations of the coordinate of the compact manifold, S ' . Put another way, 
the original five-dimensional general coordinate invariance is spontaneously broken 
1092 D Bailin and A Love 
in the ground state. In this way, we arrive at an ordinary theory of gravity in four 
dimensions, together with a theory of an Abelian gauge field, with connections between 
the parameters of the two theories because they both derive from the same initial 
five-dimensional Einstein gravity theory. 
We adopt coordinates A = 1,. . . , 5 with 
p = X f i p=o,1,2,3 (1.1) 
2 5 = e (1.2) 
being coordinates forordinary four-dimensional spacetime, and 
being an angle to parametrise the compact dimension with the geometry of a circle. 
The ground-state metric after compactification is 
E% = diag{T,, - i 5 5 ) 
77,u = (1, -1, -1, -1) 
iS5 = d2 (1.5) 
(1 .3 ) 
(1.4) 
where 
is the metric of Minkowski space, M4, and 
is the metric of the compact manifold S' , where E is the radius of the circle. 
the ground state. Quite generally, we may parametrise the metric in the form 
The identification of the gauge field arises from an expansion of the metric about 
(1.6) 
>. 
g,Jx , 6 ) - B,(x, O)Bu(x, e ) @ ( & e) B,(x, e w x , 0 ) 
w x , w x , e ) - W X , 8 ) gAB(x, e) = ( 
To extract the graviton and the Abelian gauge field it proves sufficient to replace 
@(x , 8 ) by its ground-state value i55, and to use the ansatz without 8 dependence: 
We write 
B, (x ) = 5A, (x ) (1.8) 
where 5 is a scale factor we shall choose later so that A , ( x ) is a conventionally 
normalised gauge field. 
1.3. Abelian gauge transformations 
Coordinate transformations associated with the coordinate 8 of the compact manifold 
may be interpreted as gauge transformations, as we now show. Consider the trans- 
formation 
e + e '= O + ( E ( X ) . (1.9) 
For a general coordinate transformation 
(1.10) 
Kaluza- Klein theories 1093 
For the particular transformation (1.9), the off-diagonal elements of the metric give 
A,+AL=A,+d,e . (1.11) 
Thus the transformation (1.9) of the coordinates of the compact manifold induces an 
Abelian gauge transformation on A,. This means that the compact manifold is provid- 
ing the internal symmetry space for the (Abelian) gauge group, and internal symmetry 
has now to be interpreted as just another spacetime symmetry, but associated with the 
extra spatial dimension. 
1.4. Effective four-dimensional action 
An effective action for the four-dimensional theory may be derived from the action 
for five-dimensional Einstein gravity 
(1.12) 
where is the five-dimensional curvature scalar, and C? is the gravitational constant 
for five dimensions. Substituting the ansatz (1.7) for gAB, and integrating over the 
extra spatial coordinate 8, gives an effective four-dimensional action 
I=-- 2Td / d4xldet gI”*R -- - zG [ d4x)det gl”2F,yFCLY 
1 6 ~ G 
(1.13) 
where d is the radius of the compact manifold as in (1.5), R is the four-dimensional 
curvature scalar, and 
FFY zz a,A, -&A,. (1.14) 
The four-dimensional gravitational constant G is thus identified as 
G = G / 2 ~ d . (1.15) 
To obtain standard normalisation for the gauge field we must then choose 
(1.16) 
where 
K ~ = 1 6 ~ G . (1.17) 
Then the effective four-dimensional action is 
(1.18) 
(Had the extra dimension been timelike, we would have obtained the opposite (wrong) 
sign for the Maxwell action in (1.13).) 
Whether retaining just the massless states in four dimensions in this way is an 
adequate approximation is discussed in § 2.3. 
1094 I) Bailin and A Love 
1.5. Mass eigenstates 
The natural scale of mass for these theories is the Planck mass and massive fields in 
five dimensions will naturally lead in four dimensions to particles with masses on the 
Planck scale. Suppose instead we start with a massless field in five dimensions. For 
a five-dimensional scalar field 4 ( x , e ) we may make the Fourier expansion on the 
compact manifold 
m- 
The Klein-Gordon equation 
then gives the equations for the Fourier components: 
(0, + m2, )4"(x ) = 0 
m2, = n 2 / i 2 . 
where 
(1.19) 
( 1.20) 
(1.21) 
( 1.22) 
The fields +"(x) are thus the mass eigenstates in four dimensions, and the field 4'(x) 
is the only massless one (or perhaps light, after allowing for radiative corrections). 
The other fields 4 " ( x ) have masses of order k', which we would expect to be 
comparable to the Planck mass. If the extra dimension had had a timelike signature 
(positive in our convention), we would have obtained a negative mass squared in 
(1.22), i.e. tachyons. 
1.6. Charge quantisation 
If we apply the coordinate transformation 
e+ e '= e + t E ( X ) 
4 ( X I + exp(in& ( x ) ) 4 " (XI. 
to the field +(x, 6 ) of (1.19), we have 
Since the Abelian gauge field transforms (according to (1.11)) in the manner 
(1.23) 
(1.24) 
A, + A: = A, + dP& (1.25) 
this means that 4 " ( x ) has charge 
qn = - n t = - n K / E (1.26) 
where we have used the normalisation condition (1.16). Thus, charge is quantised in 
units of tc/k. The radius of the compact manifold may now be estimated from 
I?'= K2/e2 = 4 G / ( e 2 / 4 r > . (1.27) 
Thus, identifying e with the quantum of electric charge 
I? - l o m i ' . (1.28) 
There is however a flaw in the five-dimensional Kaluza-Klein theory, even if we d o 
not wish to include weak and strong interactions as well as electromagnetism. From 
(1.26), we see that all charged particles have n # 0. But, from (1.22), this means that 
they all have masses on the Planck scale mp, whereas the familiar charged particles 
have very small masses on that scale. 
Kaluza- Klein theories 1095 
1.7. Scalar jield from metric 
Starting from the general parametrisation of the metric (1.6), it is possible to extract 
a massless scalar field (Freund 1982, Appelquist and Chodos 1983a, b). Deleting 8 
dependence, which is associated with massive degrees of freedom, the graviton-scalar 
sector may be obtained from 
( 1.29) 
Substituting in the action (1.12) for five-dimensional gravity leads to the effective 
four-dimensional action 
(1.30) 
The @ I / * multiplying the four-dimensional curvature scalar may be removed by a Weyl 
scaling, 
Then, 
or, with the further change of variables 
K - I 
d3 x = - In(@/@,) 
we have 
I=------ d4xldet gI1I2 a”x 3,x. 
1 6 r G ‘I 
(1.31) 
(1.32) 
(1.33) 
(1.34) 
This is a particular case of four-dimensional Brans-Dicke theory (Jordan 1959, Brans 
and Dicke 1961). 
2. (4 + D)-dimensional Kaluza-Klein theories 
2.1. Isometry group of a manifold 
In order to unify gravitation, not just with electromagnetism but also with weak and 
strong interactions, it is necessary to generalise the five-dimensional theory of 5 1 to 
a higher-dimensional theory (Klein 1926, DeWitt 1964, Kerner 1968, Trautman 1970, 
Cho 1975, Cho and Freund 1975, Scherk and Schwarz 1975, Cremmer and Scherk 
1977) so as to obtain a non-Abelian gauge group. In the five-dimensional case, an 
Abelian gauge group arose from the coordinate transformation 
e+ e ’ = e + t E ( X ) (2.1) 
on the single coordinate 8 of the compact manifold. In the (4+ D)-dimensional case 
we must look for symmetries of the compact manifold which generalise (2.1). The 
1096 D Bailin and A Love 
appropriate transformations to study are the isometries of the manifold (an introductory 
discussion of isometries is to be found in Weinberg 1972, ch 13). Let us denote the 
coordinates of ordinary four-dimensional space by x@, and the coordinates of the 
compact manifold K by y". An isometry of K is a coordinate transformation y + y' 
which leaves the form of the metric gm,, for K invariant: 
y - ) y ' : g ~ n ( y ' ) = ~ m , ( y ' ) . (2.2) 
Isometries form a group, with generators t, and structure constants C a b c , in the following 
way. The general infinitesimal isometry is 
I + iaata : y" -+ y "' = y" + s"t ,"(y) (2.3) 
where the infinitesimal parameters are independent of y , and the Killing vectors 
t:, which are associated with the independent infinitesimal isometries, obey the algebra 
t Y a m t Z - t Y a m t E = - C a b c t I . (2.4) 
Correspondingly by considering the commutator of two infinitesimal isometries, we 
can show that 
[ la , t b l = i C a b c f c * (2.5) 
For instance, theN-dimensional sphere S N has isometry group SO( N + l ) , and the 
2N(real)-dimensional complex projective plane CPN has isometry group SU( N + 1). 
The isometry group for the compact manifold S' of the five-dimensional theory is just 
the SO(2) (or U(1)) group of transformations of (2.1). As we shall discuss later, it is 
possible to choose the compact manifold to obtain the isometry group SU(3) x SU(2) x 
U(1), which is the (observed) gauge group of electroweak and strong interactions. 
2.2. Non-Abelian gauge transformations 
The ground-state metric for the compactified (4+ D)-dimensional theory may be written 
as 
EAB = diag"' { v f i y , -i,,(y)I (2.6) 
where vfiu is the metric of Minkowski space M4 as in (1.4), and g,,(y) is the metric 
of the compact manifold. We return in § 3 to discuss whether such a ground state 
does exist. The non-Abelian gauge fields of the theory may be displayed by the 
expansion about the ground state 
(2.7) 
with 
(A more general ansatz including x dependence for imn and y dependence for g,, and 
A, is necessary to display any massless scalars arising from the metric and massive 
states.) 
Non-Abelian gauge transformations arise by considering the effect on the com- 
ponents gpn of the metric of the infinitesimal isometry with x-dependent parameters: 
Kaluza- Klein theories 1097 
We then find that 
A: + A ; = A;+d,&"(X)+ CabcBb(X)AL (2.10) 
which is just the usual Yang-Mills gauge transformation if we display the gauge 
coupling constant g explicitly by writing 
C a b c = d a b ? (2.11) 
t a = g Ta (2.12) 
and 
so that 
[ Ta, Tb] = ifabcTc. (2.13) 
Thus, non-Abelian gauge transformations are generated by x-dependent infinitesimal 
isometries of the compact manifold K. 
2.3. Effective four-dimensional action 
The action for Einstein gravity in 41- D dimensions is 
(2.14) 
where R is the (4+ D)-dimensional curvature scalar, and e is the gravitational constant 
for 4 + D dimensions. Substituting the ansatz (2.7) for gAB, and integrating over the 
compact degrees of freedom y gives an effective four-dimensional action 
x (16&)-' - d4xldet gl'/2FLlya(FI*Y)b (2.15) 
4 'I 
(2.16) 
and R denoting the four-dimensional curvature scalar. The four-dimensional gravita- 
tional constant G is thus identified by 
K - * = (16.rrG)-'= (167rG)--' dDyldet g(y ) / ' " (2.17) J 
and standard normalisation of the gauge fields requires the Killing vectors to be scaled 
so that 
(6:(y)6:(y)gmmn(y)) = K28ab (2.18) 
where we have introduced the notation (of Weinberg 1983) 
(2.19) 
1098 D Bailin and A Love 
Then we have the standard action for Einstein gravity plus non-Abelian gauge fields 
in four dimensions: 
I = - ( 1 6 ~ G ) - ' d4xldet g1'%! -- d4xldet g11'2Fp/(FpY)a. (2.20) I 4 'I 
The discussion given above has to be modified if the higher-dimensional theory 
one starts from is supergravity rather than ordinary Einstein gravity. For instance, 
there is present in eleven-dimensional supergravity a third-rank antisymmetric tensor 
field. (See, for example, the review articles of Englert and Nicolai (1983) and Duff et 
a1 1986.) In an expansion of this field about its expectation value, terms proportional 
to the gauge fields appear (Duff et a1 1983c, 1984a) and in four dimensions there is 
an additional contribution for the Yang-Mills Lagrangian. Then, the normalisation 
of the Killing vectors in terms of the gravitational constant differs from (2.18). This 
amounts to a modification of the relationship between the gauge coupling constant 
and the gravitational constant, as can be seen from § 5 . 
A further subtlety, which has been emphasised by Duff et a1 (1984b) and Duff and 
Pope (1984), is that simply retaining the massless states in the four-dimensional theory 
after compactification may give field equations whose solutions are not exact solutions 
of the full (4 + D)-dimensional field equations. (This includes the five-dimensional 
case.) However, it has been argued by Witten and Weinberg (see Duff and Pope 1984) 
that the errors incurred are suppressed at energy scale E by powers of EIE , , where 
Eo is the compactification scale, provided the four-dimensional ground state is 
Minkowski space. 
2.4. Graviton-scalar sector 
The graviton-scalar action may be derived by making the ansatz 
(2.21) 
which is more general than (2.7), in that it allows x dependence for g",,. Substitution 
of this ansatz in the (4+ D)-dimensional gravitational action (2.14) yields (Cho and 
Freund 1975) 
I = - ( 1 6 ~ G ) - ' d4x dDy)det gp,l''21det @m,/"2 i I 
x {[w + fi - amn D, DP a,, - 1 D'* amn D , am" 
- @"'"D' 4)mn@p4Dp(Pp4 + @"n(PPYD,~pmDpL(P4,} (2.22) 
where R and Lk are the curvature scalars for 4 and D dimensions, respectively. The 
integration over the compact manifold 5 d"y may be performed given the metric. 
2.5. Diferential geometry 
The modern formulation of differential geometry, in terms of differential forms, has 
great calculational and notational advantages for studying the compact manifold of 
Kaluza-Klein theory. Accordingly, we give here a brief resumt of the more important 
definitions and results. For a more extensive and thorough discussion see, for instance, 
Eguchi et a1 (1980) and Salam and Strathdee (1982). Differential forms are a convenient 
Kaluza- Klein theories 1099 
formalism for manipulating totally antisymmetric tensors. An antisymmetric product 
of coordinate differentials (the exterior product A ) is defined so that 
(2.23) dy" A dy" = -(dyn A dy") 
and a p-form w is constructed from any pth-rank antisymmetric tensor w m I , . mp : 
~ ~ ~ ~ , , , , , , , ~ d y " ~ ~ d y " ~ ~ . . . ~ d y " p . (2.24) 
Then for a p-form ap and a q-form P, 
( Y p A P, = (-1)"'PP, A C y p . (2.25) 
A totally antisymmetric differentiation, the exterior derivative d, is defined by 
dw = a m p + l ~ m , . . , mp dy"p+l A dyml A . . . A dy".. (2.26) 
There follows immediately the important property 
ddw = O (2.27) 
(2.28) 
An important theorem which generalises Gauss's Law and Stokes's theorem is that 
r r 
(2.29) 
where M is a p-dimensional manifold and dM is its boundary. 
It is often convenient to work with the vielbein e*,(y) for the compact manifold K 
rather than its metric i m n ( y ) . If the tangent space to the manifold is a Euclidean space 
so that we may choose its metric to be ti,,, then the vielbein is defined by 
imn(Y) = aope",~)e!(~ I * (2.30) 
(We shall refer to a, P, . . . as flat indices, or tangent space indices, and m, n, . . . as 
curved indices.) The vielbein has inverse e,"(y) given by 
e,"(v) = &3i""(Y)e!(Y) (2.31) 
(2.32) 
(2.33) 
Flat indices may be converted into curved indices, and vice versa, using the vielbein 
and its inverse. 
The vielbein 1-form 
e"(y) = eXY) dY" (2.34) 
may be used to construct the spin-connection 1-form U",, and the torsion and curvature 
2-forms T" and R",: 
(2.35) de" + w e p A eo = T" 
and 
R", = dw", + w a y A w y p 
-4 Reprs(eY A e')). (2.36) 
1100 D Bailin and A Love 
(We shall normally deal with manifolds which admit a torsion-free metric, T" =O.) 
The usual curvature tensor Rmnp4 is obtained by using the vielbein to convert flat 
indices into curved indices. The relationship of the spin connection as defined by 
(2.35) to the usual definition (as, for example, in Weinberg 1972, § 12.5) may be read 
off from the covariant derivative Om$ of a field $ transforming as the representation 
M"' of the SO(D) tangent space group of the compact manifold K : 
O m $ = ( a m -iUm)$ (2.37) 
where 
w e P = (wclp),dym (2.38) 
and 
w, =f (U,'),M"'. (2.39) 
As a 1-form, 
D$ = (d - iw)& (2.40) 
When the compact manifold K is a coset space G/H, the vielbein may be constructed 
as follows. The points y" of K maybe represented by chosen elements L ( y ) of G, 
one from each coset. Let the Hermitian generators of G be Qc; with commutation 
relations 
(2.41) 
may be written in the form 
(2.43) 
where the generators Qc; have been divided into those, Q6, which belong to H, and 
the others Qa which are associated with the tangent space. Then, e"(y) is the vielbein 
1-form for G / H as in (2.34). 
The spin connection for a coset space is conveniently calculated from the structure 
constants of the group G using the Maurer-Cartan formula 
(2.44) M y ) + e(y) A e(y) = 0 
which follows from (2.42) and 
ddL(Y 1. (2.45) 
In terms of the structure constants of G this is 
dec;((y) = 1 f&( e6((y) A e'(y)). (2.46) 
In particular 
d e " ( y ) = i f a Y p ( e Y ~ e ' ) + f & ( e ' ~ e'). (2.47) 
Comparing with (2.35) for zero torsion we identify the spin connection 
Kaluza- Klein theories 1101 
In particular for the important case of a symmetric coset space where 
f a & = 0 (2.49) 
we have the simple form 
w a p = fOspes. (2.50) 
It is important in applications to know that the generators Q) of H may be embedded 
in the tangent space group SO(D) for K. The reasoning is as follows. A left translation 
y + y' may be defined on K by 
& ( Y ) = U Y ' ) h (2.51) 
where g is an arbitrary element of G, and h is a suitable element of H. When g is 
independent of position on the manifold, it may be deduced from (2.42) (Salam and 
Strathdee 1982) that under left translations 
e a ( y ' ) = e P ( y ) D p " ( h - ' ) (2.52) 
where Dop is a matrix of the adjoint representation of G, defined by 
g-'Q;g = D; p^(g)Qp.. (2.53) 
Equation (2.52) specifies an embedding of H in the tangent space group SO(D) . In 
terms of the matrix elements of the adjoint representation of G, 
( Q p I a p = -iCwp. (2.54) 
But the standard generators of S O ( D ) are 
= -i(6;6{ - 6;s;). 
Thus the embedding is 
(2.55) 
2.6. The manifolds Mpqr 
To obtain the known gauge group SU(3) x. SU(2) x U ( l ) of strong and electroweak 
interactions as (a subgroup of) the isometry group of the compact manifold K , it is 
necessary to have K at least seven dimensional. This is because (Witten 1981) the 
lowest-dimensional manifold with isometry group G is a coset space G / H with H a 
maximal subgroup of G, but with none of the factors in H identical to any factor in 
G. Thus, for 
(2.57) G = SU(3) x SU(2) x U( 1) 
we must choose the maximal subgroup 
H = SU(2) x U'( 1) x U''( 1) (2.58) 
leading to the manifold 
K = G / H (2.59) 
of dimensionality 12 - 5 = 7 . 
Almost the most general coset space of this type, denoted MP9' by Witten, may be 
constructed as follows. Denote the generators of SU(3), SU(2) and U ( l ) by $ A o , 
a = 1, . . . ,8 , fa,, (Y = 1,2,3, and Y. It is necessary (without loss of generality) to select 
1102 D Bailin and A Love 
two U( l ) factors commuting with the SU(2) isospin subgroup of SU(3) to be in H. In 
other words, it is necessary to select one combination of $ i 8 , tu3 and Y which does 
not occur in H (and so, as in § 2.5, is associated with the tangent space). Let this 
combination be 
2 E $(&PA8 + qU3 + 2 r Y) (2.60) 
with p , q and r arbitrary integers to give a compact U( 1). Then, the two combinations 
which lie in H may be taken to be the orthogonal combinations 
Z ’ = $ [ 2 & p r A 8 + 2 q r u 3 - 2 ( 3 p 2 + q 2 ) Y ] (2.61) 
and 
2’’ = f( -&qA 8 + 3pU3). (2.62) 
This completes the construction of K . 
Generally, the isometry group for MP4‘ is SU(3) x SU(2) x U(1). However, for the 
exceptional cases ( p = 1, q = 0, r = 1) and ( p = 0, q = 1, r = 1) the manifolds are S5 x S2 
and CP2 x S3 with the larger isometry groups SO(6) x SO(3) and SU(3) x S0(4) , respec- 
tively (see § 2.1). 
The manifolds Mpqr are not quite the most general (orientable) seven-dimensional 
manifolds with isometry group at least SU(3) x SU(2) x U( l), because it is not necessary 
to chose 2,Z’ and 2” orthogonal. All that is required is that they should be independent 
(Randjbar-Daemi et a1 1984a). Thus, we may choose 2 as in (2.60), but take the two 
U(1) factors in H to be 
X’=f&A8+sY (2.63) 
and 
X”=i u,+tY (2.64) 
where s and t are free parameters, subject only to the constraint 
p s + qt - r # 0. (2.65) 
This slightly more general class of manifolds with isometry group SU(3) x SU(2) x U( 1) 
is labelled Mpqrsr by Randjbar-Daemi et a1 (1984a). 
3. Compactification mechanisms 
3.1. General considerations 
For a Kaluza-Klein theory to be able to describe the observed four-dimensional world 
it is necessary for the extra spatial dimensions to be compactified (except possibly in 
the early universe) down to a size which we do not probe in particle physics experiments 
(e.g. the Planck length). As has been emphasised by Appelquist and Chodos (1983a, b), 
a basic difference between five-dimensional Kaluza-Klein theory and (4 + 
D)-dimensional theory (with D > 1) is that the five-dimensional gravitational field 
equations have a compactified classical solution of the form (1.3), but, in general, the 
(4+ D)-dimensional equations do not have a compactified classical solution of the 
form (2.6). Thus, in the (4+ D)-dimensional theory, compactification of the D extra 
spatial dimensions requires that either matter fields are introduced to provide an 
energy-momentum tensor, or that the (4+ D)-dimensional gravitational action differs 
Kaluza- Klein theories 1103 
from the minimal Einstein action. We shall mostly discuss the former possibility, but 
return to the latter possibility in 8 3.8. 
When compactification is due to matter fields, an energy-momentum tensor .TAB 
must be introduced on the right-hand side of the (4+ D)-dimensional gravitational 
field equations: 
RAB -$(R + ~ ) E A B = ~ T ~ T A B (3.1) 
where for generality a (4 + D)-dimensional cosmological constant has been included. 
(The following argument follows closely Randjbar-Daemi et a1 (1983b).) If we demand 
compactification into M4 x K , where M4 is four-dimensional Minkowski space, and K 
is the D-dimensional compact manifold, then 
- 
R,, = 0 p , v = o , 1 , 2 , 3 (3.2) 
and because of Lorentz invariance the 4-space components of the energy-momentum 
tensor are of the form 
(3.3) 
where r lPu is the Minkowski metric of (1.4). If the compact manifold is an Einstein 
space we may also write 
- I 
Rmn = -2kEmn k">O (3.4) 
and the components of the energy-momentum tensor on the compact manifold are of 
the form 
It then follows from the field equations (3.1) that 
iw,, =(E-c )& , 
iw = g m n R mn = D ( E - c) 
x = -DE+ ( D - 2)c. 
and 
Equation (3.7) shows that compactification occurs provided 
E - c < O . 
(3.8) 
(3.9) 
3.2. Freund-Rubin compactifcation 
A mechanism for compactification which arises naturally in eleven-dimensional super- 
gravity (see P 7.2) has been particularly explored by Freund and Rubin (1980). This 
mechanism depends on a third-rank antisymmetric tensor field A B C D with field strength 
FABCD a A A s c D - ~ ~ A A C D + ~ C A A B D - ~ D A A B C (3.10) 
and action 
r 
r= d 4 + D ~ l d e t El"*(-& FABCDFABCD) J (3.11) 
1104 U Bailin and A Love 
(apart from couplings to fermions, and trilinear self-couplings which do not contribute 
for the type of solution considered here). Then, the field equation for F A B C D is 
(3.12) Id et gl - -'/2aA(ldet g11/2FABCD) = 0 
and the energy-momentum tensor is 
(3.13) 
The field equation (3.12) has a solution of the type 
F+W"=ldet gl - l /2EPvP"F P, v, P, u=o, 1,293 (3.14) 
and all other entries zero, where det g refers to four-dimensional space, F is a constant 
and the Levi-Civita symbol is defined such that 
EO123 = 1, (3.15) 
For such a solution the energy-momentumtensor takes the form 
F2 d e t g - 7 =--- 
P 2 /det gl gpb 
and 
- F 2 d e t g - T =-- 
*' 2 ldetgl gmn 
(3.16) 
(3.17) 
where m, n run over the compact manifold. If we look for a compactification with 
Minkowski four-dimensional space, then in the notation of (3.3) and (3.5), 
(3.18) 
Thus (3.9) is satisfied and compactification occurs. From (3.8), consistency requires 
a (4 + D)-dimensional cosmological constant 
A = 8 d ( D - 1)F2. (3.19) 
However, in the case of eleven-dimensional supergravity, an eleven-dimensional 
cosmological constant in the theory destroys the supersymmetry. In that case, one 
must set to zero, and one then finds 
iw = 8 v C F 2 4 ( D - 1)/(D+ 2) (3.20) 
and 
d = -8vC?F23D/(D+2) (3.21) 
where R and k denote the four- and D-dimensional curvature scalars, respectively. 
Thus, for eleven-dimensional supergravity, there is the unwelcome feature of an anti- 
de Sitter four-dimensional space! 
3.3. Quantum fluctuations in massless higher-dimensional $fields 
An alternative compactification mechanism (Candelas and Weinberg 1984, Weinberg 
1983) is through quantum fluctuations of massless fields in the (4+ D)-dimensional 
theory. As these authors observe, for it to make sense to consider quantum fluctuations 
in light matter fields, while neglecting gravitational quantum connections, it is necessary 
Kaluza- Klein theories 1105 
for the number of light matter fields to be large. By dimensional analysis, when the 
matter fields are massless in (4+ D)-dimensions, we may write the matter field action 
as 
I= - d4xlgl"* V( d ) (3.22) I 
where d is the radius of the compact manifold and 
v( d ) = CD2k4 (3.23) 
with CD a constant dependent on the number of matter fields. The energy-momentum 
tensor may be computed using 
(3.24) 
RD dDyldet g'l"' J 
Also, for an Einstein space, 
(3.27) 
(3.28) 
with k">O for a compact manifold. Demanding that four-dimensional space be 
Minkowski, the field equations (3.1) yield 
and with V ( d ) as in (3.23), 
(3.29) 
(3.30) 
(3.31) 
where 
G = (3.32) 
as in (2.17). 
It can be seen from (3.30) that for compactification to occur CD must be positive. 
(Alternatively, this follows from (3.25), (3.26) and (3.9).) For any given manifold, Cr, 
may be evaluated by calculating the eigenvalues of the Klein-Gordon or Dirac operator 
on the compact manifold (for massless scalar or spinor fields in 4 + D dimensions, 
respectively) to obtain the masses of the 'tower' of four-dimensional fields (see $0 1.5 
1106 D Bailin and A Love 
and 5.1), and then calculating the effective potential in the usual way (Coleman and 
Weinberg 1973). 
A similar discussion has been given earlier for five-dimensional Kaluza-Klein 
theory (Appelquist and Chodos 1983a, b). In this case k' is zero, because SI is flat, 
and no solution for d arises. The coefficient CD is found to be negative, and the 
energy V ( d ) decreases without limit as d approaches zero. Thus, it has to be assumed 
that as I?: approaches the Planck length, and the loop expansion of the effective potential 
fails, the dynamics stabilise d at that scale. This is in contrast to the situation in the 
absence of quantum corrections where any value of R satisfies the classical field 
equations (neutral stability). 
3.4. CompactiJcation due to explicit gauge jields 
It is contrary to the spirit of the original Kaluza-Klein theory to introduce gauge fields 
explicitly in 4 1 0 dimensions, since the hope was that all gauge fields might arise 
from the isometry group of the manifold, upon dimensional reduction. However, there 
are three advantages to be derived from doing so. Firstly, the explicit gauge fields 
provide a compactification mechanism. Secondly, the D 'extra' components of the 
(4+ D)-dimensional gauge fields provide scalar fields in four dimensions which may 
prove useful as Higgs scalars. The F+yFILY terms in the (4+D)-dimensional gauge 
field kinetic term provide in four dimensions the gauge field kinetic term, the FILmFpm 
terms give the Higgs scalar kinetic terms, and the F,,,,F"" terms provide the Higgs 
scalar potential, after integration over the compact manifold. (This possibility has 
been advocated by Manton (1979), Forgacs and Manton (1980) and Chapline and 
Manton (1981), though with a mathematical dimensional reduction rather than a 
physical compactification.) Thirdly, an explicit gauge field expectation value in a 
topologically non-trivial configuration can overcome the difficulty endemic to pure 
Kaluza-Klein theories of obtaining chiral fermions in four dimensions (see Q 5). 
Kaluza-Klein theories with explicit gauge fields (higher-dimensional Einstein-Yang- 
Mills theories or supergravity Yang-Mills theories) have been explored by Cremmer 
and Scherk (1976, 1977), Horvath et a1 (1977), Horvath and Palla (1978), Luciani 
(1978), Randjbar-Daemi and Percacci (1982), Randjbar-Daemi et a1 (1983a, b, c), 
Witten (1983), and many subsequent authors, the most recent theories of this type to 
excite interest being the ten-dimensional supergravity Yang-Mills theories derived 
from superstring theory (Green and Schwarz 1984, Gross et a1 1985). 
For explicit gauge fields with field strength F t B , where A and B run over all 4+ D 
dimensions, the action is 
(3.33) 
and the energy-momentum tensor is 
T A B = - ( ( F a ) AC ( F a ) B C - $ ( F a )CDFaCDgAB 1. (3.34) 
Assuming that the field strength has a non-zero expectation value only on the compact 
manifold, i.e. for A, B, C, 0, . . . taking values m, n, p, 4 , . . . , and that the compact 
manifold is an Einstein space, then in the notation of Q 3.1 we have 
(3.35) 
Kaluza- Klein theories 
and 
1107 
(3.36) 
Thus, (3.9) is satisfied consistently with compactification occurring. 
3.5. Monopole solutions 
The simplest example of compactification using a topologically non-trivial explicit 
gauge field configuration (Randjbar-Daemi et a1 1983a) is obtained for D = 2 with 
compact manifold the surface of an ordinary sphere: 
S2=G/H=SU(2) /U(1) . (3.37) 
Then an explicit U ( l ) gauge field may be introduced with a monopole expectation 
value on the compact manifold: 
(3.38) 
where a is a constant to be determined in terms of the charges of the matter fields, 
and the plus and minus signs refer to the upper and lower hemispheres, respec- 
tively. The monopole solution is SU(2) invariant (up to a U ( l ) gauge transformation) 
(Randjbar-Daemi et a1 1983a). 
The action for this theory is 
A , ( y ) dy" = a( 1 'F cos e ) d 4 
(3.39) 
and the requirement that (3.38) is a solution of the field equations, for Minkowski 
four-dimensional space, determines the radius of the compact manifold in terms of 
the six-dimensional gravitational constant and the field strength a: 
i'= 8.rrGa2. (3.40) 
When both the metric and the U ( l ) gauge field are expanded about the ground 
state the situation is more complicated than for pure Kaluza-Klein theory. In the pure 
Kaluza-Klein theory of 3 2, the SU(2) isometry group of S2 leads to SU(2) gauge fields 
in four dimensions arising from the off-diagonal components of the metric. In the 
present case, the gauge group of the effective four-dimensional action is SU(2) x U( l) , 
with the SU(2) gauge fields being a superposition of the gauge fields from the metric 
and the original explicit U( 1) gauge field. Because the radius of the compact manifold 
is related to the gravitational constant and the Yang-Mills field strength by (3.40), the 
gauge coupling constant g for the four-dimensional SU(2) gauge group is related to 
the monopole strength 
g 2 = 3/2a2. (3.41) 
3.6. Instanton solutions 
If, instead, the compact manifold is 
S4= G / H = SO(S)/S0(4) (3.42) 
then explicit SU(2) gauge fields may be introduced and an instantonsolution on the 
compact manifold may be used (Randjbar-Daemi et a1 1983c) instead of a monopole 
1108 D Bailin and A Love 
solution. (This solution is only formally an instanton. It is constructed using four 
spatial coordinates, rather than three spatial coordinates and a time coordinate. It is 
invariant, up to a gauge transformation under the action of SO(5) on the manifold.) 
Then, an expansion of the metric and the explicit gauge fields about the ground state 
leads to SO(5) Yang-Mills fields in four dimensions (rather than SO(5) x SU(2)), 
because the original SU(2) gauge symmetry is spontaneously broken by the instanton 
solution. 
3.7. Generalised monopole and instanton solutions 
If the compact manifold is the coset space 
K = G / H (3.43) 
then, in the notation of 0 2.5, a G-invariant solution of the Einstein-Yang-Mills field 
equations may always be obtained (Randjbar-Daemi and Percacci 1982) by taking 
A" G A: dy" = e" H # U(1) (3.44) 
or 
A" = ae" H = U ( l ) (3.45) 
where a is a constant, and e" are the components ofthe covariant basis (2.43) associated 
with the generators of H. In (3.44) and (3.45), it is understood that an embedding of 
H in the explicit Yang-Mills group GYM has been specified (which of course requires 
that GYM is large enough to contain H). The gauge field configuration of (3.44) or 
(3.45) generalises the monopole and instanton solutions of the last two sections to an 
arbitrary coset space. If H is of the form 
H = H'OU( 1 ) (3.46) 
then a monopole solution may be constructed by applying the ansatz (3.45) to the 
U( 1 ) factor. For instance, monopole solutions have been employed by Watamura 
(1983, 1984) for the complex projective planes 
S U ( N + l ) CP = ? - S U ( N ) x U ( 1 ) . (3.47) 
If H is of the form 
H = H'OSU(2) 
then an instanton solution may be constructed by applying the ansatz (3.44) to the 
SU(2) factor displayed. (In § 3.6, H' was SU(2).) 
Such gauge field configurations are in general topologically non-trivial. For 
instance, if for a monopole solution we write 
F = Fm,dym A dy" (3.48) 
(with the gauge coupling absorbed in the definition of the gauge field) then the first 
Chern number (Eguchi et a1 1980) 
F = integer (3 -49) 
is the monopole number, e.g. 
Kaluza- Klein theories 1109 
c, = -2a for S2 (3 .50) 
with a as in (3.38). 
For an instanton solution on S4, if we write 
Y" AdY") 
tu FGn F E -- -(d 
2 2 
(3.51) 
where ( tu/2) are the matrices representing the S U ( 2 ) factor of H in the fundamental 
representation of GYM, the second Chern number (Eguchi et a1 1980) 
1 
C2 = - Tr(F A F ) = integer 
8 r 2 K 
(3.52) 
is the instanton number, and for an instanton solution 
Ic21 = 1. (3.53) 
(There is no contribution from the curvature to C, for S4.) 
Wilczek 1977, Witten 1983) is 
Another very natural choice of gauge field configuration (Charap and Duff 1977, 
A"P Amn' dy'" = "ma' dy" I ma' (3.54) 
where umaP is the spin connection for K of $ 2.5, and the SO(D) tangent space group 
has been embedded in GYM. (A"' is a labelling of the gauge fields corresponding to 
the standard antisymmetric generators M a p of SO(D).) Then the topology of the 
gauge field configuration is directly related to the topology of the manifold (the Euler 
characteristic). 
3.8. Non-minimal gravitational actions 
A possible compactification mechanism which does not require the introduction of 
matter fields is through non-minimal terms added to the ( 4 + D)-dimensional Einstein 
action. Wetterich (1982a) has considered an action of the form 
f = -( 1 6 ~ & ) - ' 5 d4iDZldet + aR2+ / 3 R A B R A B + Y R A B C D R ~ ~ ~ ~ } . (3.55) 
He finds that the field equations admit a compactified solution of the type M 4 x S D , 
where M4 is Minkowski space, though to obtain other compact manifolds (e.g. a 
product of spheres) requires higher curvature invariants to stabilise the effective action. 
3.9. Stability of compactijication of pure Kaluza-Klein theories 
The simplest perturbation under which to consider stability (Candelas and Weinberg 
1984) is a change in the overall scale k of the manifold. Suppose that the action can 
be written in the form 
f = - ( 1 6 v G ) - ' 1 d4+'D3"l'/2(Il%(y)+~) - 5 d4xX(gl'/'V(k) (3.56) 
where k(y) is the curvature scalar for the compact manifold, and V ( k ) is a compactify- 
ing action. Using (3.28) for an Einstein space, 
fi = -2D&-'. (3.57) 
1110 D Bailin and A Love 
Integrating over the coordinates y of the compact manifold, and using (3.27) and 
(3.32), we may write (for Minkowski four-dimensional space) 
I = - d% v,,, (3.58) I 
with 
Veff= ( 1 6 ~ G ) - ' ( A - 2 b k 2 ) + V( I?). (3.59) 
When V(I?) has the simple form 
V ( d ) = c,d-4 with C D > o (3.60) 
the effective potential always has a stable minimum, when k" # 0, provided 
q + D - 2 > 0. (3.61) 
This includes the case of compactification by quantum fluctuations in matter fields 
(Candelas and Weinberg 1984), when q = 4. It also includes (Bailin et a1 1984) the 
case of Freund-Rubin compactification because the field equation (3.12) implies that 
F of (3.14) has the property 
F - i - D (3.62) 
so that (3.16) and (3.17) imply that V ( E ) is of the form (3.60) with 
q = D Freund-Rubin compactification. (3.63) 
One subtlety which arises for compactification by matter field quantum fluctuations 
is that the (dk/dt)-dependent terms in the effective action may as a result of quantum 
effects (Gilbert et a1 1984) have a different sign from the tree approximation value 
deriving from l k (y ) . This occurs (Gilbert and McClain 1984) when the ratio of the 
number of massless scalar fields to the number of massless spinor fields in the 
higher-dimensional theory is sufficiently small. Then, instability would arise at the 
apparently stable minimum of the effective potential ! 
More general perturbations may be considered than just an overall change of scale 
of the compact manifold. For instance, for a product compact manifold and Freund- 
Rubin compactification instability occurs under independent changes of scale of the 
two spaces in the product (Duff et a1 1984b, Bailin and Love 1985a) with the radius 
of one growing at the expense of the other. A similar phenomenon occurs as a result 
of one-loop quantum corrections for a toroidal compact manifold TD, with one of the 
D dimensions contracting while the others expand (Appelquist et a1 1983, Inami and 
Yasuda 1983). Stability under more general perturbations including squashing has 
also been studied for spheres (Page 1984) and other manifolds. 
All the above considerations of stability are in terms of classical perturbations of 
an effective action. However, in the five-dimensional theory there is an instability of 
the Kaluza-Klein ground state (Witten 1982, Kogan et a1 1983) due to quantum 
tunnelling. Fortunately, there does not appear to be a (simple) generalisation of this 
phenomenon to higher dimensions (Young 1984). 
3.1 0. Stability of compactijication for Einstein- Yang-Mills theories 
When the theory contains explicit gauge fields, there is a further danger of instability 
of the compactification, because of classical perturbations in the gauge fields. For the 
Kaluza- Klein theories 1111 
case of an explicit U(1) gauge field in a monopole configuration on S' it has been 
shown by Randjbar-Daemi et a1 (1983a) that no instability arises in this way. However, 
for any non-Abelian gauge group GYM with an SU(2) invariant vacuum configuration 
on S 2 instability occurs (Randjbar-Daemi et a1 1983b) as can be seen by expanding 
to quadratic order in the classical perturbation, in the action, and looking for tachyons. 
In general (Randjbar-Daemi et a1 1983b, Schellekens 1985a, b), for a compact 
manifold G/H, only the perturbationsin the gauge bosons associated with H mix with 
the perturbations in the metric. This sector has been calculated (Schellekens 1984) for 
a general hypersphere SD, for a general explicit gauge group GYM. Except for the 
case D = 3 , there are never any tachyons in this sector, and the compactification is 
stable in this respect. 
However, even when the perturbations in the H gauge bosons do not lead to 
instability, the perturbations in the gauge bosons in G but not in H often do (Randjbar- 
Daemi 1983c, 1984c, Frampton et a1 1984a, Schellekens 1985a), as in the case discussed 
by Randjbar-Daemi et a1 (1983b). A general discussion of this type of instability has 
been given by Schellekens (1985b) for a general hypersphere S D with the explicit 
gauge fields of gauge group GYM in a generalised monopole configuration (as in 0 3.7). 
4. Gauge coupling constants 
4.1. Geometric interpretation 
In the five-dimensional case of 0 1, where the gauge group was Abelian, charge was 
quantised and the quantum of charge was related to the radius of the compact manifold 
S' (and the four-dimensional gravitational constant) as in (1.26). In higher-dimensional 
theories in six or more dimensions where the gauge group is non-Abelian we would 
like to be able, in a similar fashion, to relate the gauge coupling constant (or coupling 
constants) to the geometry of the compact manifold. The connection can be made 
(Weinberg 1983) by considering isometric curves on the manifold (i.e. curves traced 
out when an infinitesimal isometry is exponentiated). Consider the infinitesimal 
isometry with parameter d a associated with a particular generator t, of the isometry 
group. From (2.3) this is 
I + idat, : y" -+ y" ' = y" + da(:(y) (4.1) 
where 6:: is the corresponding Killing vector. The exponentiated curve traced out as 
a varies is 
where Y" is the solution of 
d Y " / d a = ( z ( y ) Y " ( a = O ) = y , " . (4.3) 
For any representation of the non-Abelian isometry group, the eigenvalues of a diagonal 
generator t, will be integral multiples of some lowest (positive) eigenvalue gmi, (with 
the gauge coupling constant g absorbed in the definition of the generators). As a 
increases from 0 to 2 r / g m j n , e'"'" returns to its starting value, and, if the representation 
t, is N valued, we go exactly N times round the manifold. The circumference S(yo ) 
1112 D Bailin and A Love 
of the manifold along the isometric curve with starting point yo is 
(4.4) 
where the metric g",,(y) is as in (2.6), and infinitesimal distance ds along the isometric 
curve is given by 
Averaging over the starting points yo , and using the notation (2.19), 
(s ' ) = (2.rr/Ng,i")2(immn(y)50m(y)5~(y)). (4.6) 
(s') = ( ~ T / N ~ , ~ , , ) ~ K ' (4.7) 
For standard normalisation of the gauge fields (2.18), 
where K is the four-dimensional gravitation constant of (2.17). Thus, the gauge coupling 
gmin is related to the geometry of the compact manifold by 
gmin = 2.rr~/N(s')'" (4.8) 
where (s ' ) ' '~ is the root-mean-square circumference of the manifold along the isometric 
curve associated with the generator t , averaged over starting points. The result is 
general enough to handle situations where the isometry group is a product of non- 
Abelian factors, so that there might be different gauge coupling constants associated 
with different diagonal generators t , . Clearly, the above reasoning is not directly 
applicable to cases where the gauge group has an Abelian factor, since Abelian gauge 
fields do not have any self-coupling. In that case, to interpret the result (Weinberg 
1983) it is necessary to introduce a matter field (e.g. a complex scalar field). 
For an Einstein space, where the Ricci tensor is proportional to the metric, a 'radius' 
a for the compact manifold may be introduced by writing 
d,, = a-'gmn. (4.9) 
For the hypersphere SD, (4.8) yields (Weinberg 1983) 
K 2 (D+1) 
g2'2 2 ( D - 1 ) (4.10) 
where the gauge coupling constant g for the SO(D + 1) isometry group has been 
normalised so that in the single-valued defining representation the eigenvalues of, say, 
MI2 are (g, -g, O , O , , . .), with the normalisation of generators 
[ M ij, M k / 1 = i( 6 i k n / r ' / + 6j'M ik - 8 i/Mjk - 6 j k M (4.11) 
and couplings M"At;,+, . . . For the four-dimensional manifold CP2, the correspond- 
ing result (Weinberg 1983) is 
g"; K = / a = (4.12) 
with a standard normalisation of generators for the isometry group SU(3), such that 
the eigenvalues of f3 in the 3 are (g/2, -g/2,0). 
Kaluza- Klein theories 1113 
4.2. Absolute values of gauge coupling constants 
When compactification is by quantum fluctuations in massless higher-dimensional 
fields, it is possible, as discussed in § 3.3, to determine the constant CD, in the effective 
potential of (3.23), by calculating with the tower of four-dimensional fields deriving 
from the harmonic expansion of the higher-dimensional fields. Then, the radius of 
the compact manifold is determined by (3.30) where, in the present notation, 
a-'= 2l3-2. (4.13) 
If for instance, the compact manifold is a sphere SD, then 
2 C = D - 1 for S D. (4.14) 
For this case, the absolute value of the gauge coupling constant (at the compactification 
scale) is then determined by (4.10). Unfortunately (Candelas and Weinberg 1984), 
very large numbers (> lo3) of massless higher-dimensional fields seem to be required 
to obtain g2/4.irs 1, as we might expect if g 2 varies under the renormalisation group 
by less than an order of magnitude between the electroweak scale and the compac- 
tification scale. 
4.3. Ratios of gauge coupling constants 
When the compactification is be some mechanism other than quantum fluctuations, it 
may not be possible to calculate absolute values of gauge coupling constants, e.g., for 
Freund-Rubin compactification (Freund and Rubin 1980) the expectation value (3.14) 
of the antisymmetric tensor field strength is unknown, and consequently the 'radius' 
of the compact manifold is unknown. However, it is still possible to calculate ratios 
of gauge coupling constants provided the geometry of the compact manifold is com- 
pletely determined apart from the overall scale. This is the case for Freund-Rubin 
compactification, because (3.17) (together with the Einstein field equation) implies 
that the compact manifold is an Einstein space, so that there is only a single overall 
scale, and the various circumferences of the manifold are all related. (A subtlety arises 
for the Kaluza-Klein supergravity case to which we return shortly.) 
For the manifolds Mpqr of § 2.6, with isometry group SU(3) x SU(2) x U(l ) (except 
in the exceptional cases mentioned in 8 2.6), the geometry is specified by three scales, 
a, b and c (Castellani et al 1984a). However, when the condition is imposed that Mp4' 
should be an Einstein space for Freund-Rubin compactification, these scales are related 
(Castellani et a1 1984a). Thus, the coupling constants g , , g, and g , for the SU(3), 
SU(2) and U( l ) factors of the isometry group are related by using (4.8), even though 
the absolute values cannot be calculated. It is found (Bailin and Love 1984b, Ezawa 
and Koh 1984a) that for all non-zero values of p, q and r 
1 < g : / g : < i (4.15) 
and 
7 ( p 2 n 2 / r 2 ) < g : / g : < a (4.16) 
where n is an integer. Equation (4.15) gives results which can be in reasonable 
agreement with extrapolations (Bailin and Love 1984a) of the known gauge coupling 
constants to the compactification scale. However, (4.16) gives values of g : / g : that are 
far too large unless r is bigger than 1, corresponding to non-simply connected manifolds 
(Witten 1981). The situation is a little better for the exceptional case S5 x S2 (Bailin 
and Love 1984a), but this product manifold is probably unstable, as discussed in 93.9. 
1114 D Bailin and A Love 
U p to this point, we have been relating gauge coupling constants to the geometry 
by assuming that the gauge field terms in the effective four-dimensional theory arise 
entirely from the (4+ D)-dimensional curvature scalar, as in 0 2.3. However, as we 
have already mentioned in 5 2.3, some refinement is necessary in the supergravity 
context where there are other terms in the (4+ D)-dimensional action which may 
contribute to the gauge field action in four dimensions. For instance, in eleven- 
dimensional supergravity (see § 7.1) contributions to the Yang-Mills Lagrangian in 
four dimensions can come from the fourth-rank antisymmetric field strength (employed 
in the mechanism of Freund and Rubin (1980)). These contributions (Duff et al 1983c) 
arise by taking an expectation value for the tensor field strength of the form (3.14), 
and writing, in the expansion about the ground state, 
F,,,, cc EpupmDPA:Dm ( inp (Y 15: ( Y 1 ) p , v = 0 , 1 , 2 , 3 , m , n = 4 , . . . , 10 (4.17) 
in the notation of § 2.2. When (4.17) is substituted in the action (7.11), extra contribu- 
tions to the gauge field kinetic term arise from DPA: in (4.17), resulting in this term 
being multiplied by a factor of four relative to the pure eleven-dimensional Einstein 
gravity case. The only effect is on the overall normalisation of the gauge field term in 
four dimensions compared with the four-dimensional Einstein gravity term, so that 
ratios of gauge coupling constants are unmodified. 
The situation can be different in other supergravity theories. For instance, in N = 2 
(two supersymmetry generators) supergravity in ten dimensions, the extra contributions 
to the Yang-Mills Lagrangian in four dimensions come from both a fourth-rank tensor 
field strength and a second-rank tensor field strength belonging to the supergravity 
multiplet, and these contributions have a diferent structure to the contribution due to 
the metric of 0 2. Then the ratios of gauge coupling constants differ from the ratios 
in pure (4+ D)-dimensional Einstein gravity. As we have seen in § 3.5, a similar 
phenomenon can occur in Einstein-Yang-Mills theories, with mixing between explicit 
higher-dimensional gauge fields and gauge fields arising from the metric on dimensional 
reduction. 
In eleven-dimensional supergravity, the ratio of the gauge coupling constants g, 
and g, for the isometry group SO(5) x SO(3) of the squashed 7-sphere (see § 7.2) has 
been calculated by de Alwis et a1 (1985) to be 
s:/s: = %. (4.18) 
5. Particle spectrum of Kaluza-Klein theories 
5.1. Boson spectrum 
We have seen in § 1.5 that fields which depend upon the coordinate(s) associated with 
the compactified dimension(s) have a mass whose scale is set by the size of the compact 
space; the fields 4 " ( x ) in (1.19), for example, have a mass 
* 
m, = n / R (n =0, 1 ,2 , . . .) (5.1) 
where is the 'radius' of the circle formed by the compact dimension. This scale is 
likely to be gigantic compared with the energies available to present or forseeable 
particle accelerators, since we saw in (1.28) that d-' is of order of the Planck mass: 
1019 GeV. (5.2) E-' - mp G-I/2- 
Kaluza- Klein theories 1115 
Nevertheless these massive states may still affect the low-energy sector (Duff 1984) 
because of quantum effects. This low-energy dependence upon the high-mass states 
derives from the propagators of the massive states 
(5.3) 
Since m, +CO as n + 00 in (5.1), we might suppose that at most it would be necessary 
to retain only the lowest excitations, say n = 0, 1,2,3, at least i n j n i t e diagrams. (There 
is also the possibility of logarithmic dependence upon m, which arises on renormalisa- 
tion of divergence diagrams.) Quite apart from the question of the renormalisability 
of Kaluza-Klein theories, the above argument for retaining at most the lowest excita- 
tions is complicated by the fact that the charge q, of the particle of mass m, also 
increases with n, as is apparent from (1.26). Since 
q, = n K / E 
= O( 16.rrmi2) as n+co. 
Thus the contribution from each component of the whole ‘tower’ of massive states is 
equally important (or unimportant). Our chief preoccupation in the succeeding parts 
of this section will be the characterisation of the zero modes of Kaluza-Klein theories, 
since it is these modes, which develop non-zero masses at a scale well below the Planck 
scale, which are presumed to constitute the particles actually observed in the (low- 
energy) world which is accessible to experiment. Even so, for the reasons already 
given, it is of some interest to characterise the massive modes, and in any case these 
modes are as important as the zero modes in determining the cosmological evolution 
of the very early universe, near the compactification scale. (We shall discuss this 
cosmological role of the massive modes in the following section.) 
We start by determining the four-dimensional classical mass spectrum of the original 
five-dimensional Kaluza-Klein theory. In the absence of any matter fields, the 
equations of motion are 
R A , - $ g A B R = o (5.5a) 
or equivalently 
- 
i W A B = O (5.5b) 
and the ground-state solution is 
( O l g A B J O ) = v A , diag(1, -1, -1, -1, -E’) (5.6) 
as in (1.3). To find the mass spectrum we vary the field gAB around its ground-state 
value. Thus we write 
g A B ( x , 6 ) = T A B + h A B ( X , 6 ) 
and expand (5.5b) to lowest- (first-) order in h. This gives 
(5.7) 
1116 D Bailin and A Love 
Note that the connection coefficients vanish in the ground state, so ordinary partial 
derivatives are sufficient. Equation (5.8) is invariant with respect to the gauge transfor- 
mation 
hAB+ ~ A B + ~ A S B + ~ ~ A (5.9) 
so we may choose a convenient gauge in which to extract the physical content of the 
theory. The choice is (Dolan 1983) 
a”hW5 = 0 ( 5 . 1 0 ~ ) 
a5hW5 = 0 (5.10b) 
a5h55 = 0. (5.1 Oc) 
Since the compactified manifold is a circle we may write 
(5.11) 
as in (1.19), so that h is well defined on SI. The gauge choice (5.10) then implies that 
d”h$(x ) = 0 ( 5 . 1 2 ~ ) 
h F i ( x ) = 0 ( n f 0) (5.12 b) 
h $ ; ’ ( x ) = 0 ( n # 0 ) . (5 .12~) 
In other words, by an appropriate choice of gauge the n # 0 vector potentials h f ? ( x ) 
and the n # 0 scalar fields h $ ; ’ ( x ) may be transformed to zero (i.e. ‘gauged away’). 
This sounds reminiscent of the situation in electroweak theory, for example, where 
the local SU(2) x U( 1) gauge invariance is spontaneously broken. The would-be 
Goldstone boson modes (which can also be gauged away) associated with spontaneous 
breakdown of the global symmetry are ‘eaten’ by the hitherto massless gauge bosons. 
In the, process some of the gauge bosons become massive. We might wonder, therefore, 
whether the n # 0 vector potentials and scalar fields are ‘eaten’ by the n # 0 tensor 
modes h F ? ( x ) , which thereby become massive. (Note that, if the n # 0 vector potentials 
and scalar fields really are (massless) Goldstone modes, then there are a total of three 
modes for each n available for eating, since a massless vector field has two degrees 
of freedom, and a massless scalar only one. Thus there would be a total of five degrees 
of freedom available, just the number required by a massive spin-2 field.) In fact, the 
scenario envisaged above is what actually happens, although it is not apparent at this 
stage precisely what symmetry it is which will yield a (three-times) infinite number 
(all n # 0) o f Goldstone modes. This is the subject of the following section. For the 
present we shall content ourselves with verifying that the n # 0 tensor fields h f J ( x ) 
are indeed massive, as we have claimed, and that all the other modes are massless 
(Salam and Strathdee 1982). The derivation whichwe present follows that of Dolan 
(1983). 
First we substitute (5.11) into (5.8), and use the gauge choice (5.10), or equivalently 
(5.11). From the 55 component of (5.8) we find 
d”a,hio,’ = 0 ( 5 . 1 3 ~ ) 
and 
hk‘“’ = 0 ( n # O ) . (5.13 b) 
Kaluza- Klein theories 1117 
The p5 component gives 
aA,j A h(O)=O l.r5 ( 5 . 1 4 ~ ) 
and, using (5.13b), 
aAht’ = 0 ( n # 0). (5.146) 
Finally, from the pv component of (5 .8 ) , we find 
aAa,h‘,Ot+a,a.(h”,o)+h:(o))-d,a I I ” hA(0)-a A U P a hA(O)=O ( 5 . 1 5 ~ ) 
and using (5.13b) and (5.14b) 
(ahar + n2/k2) hl”y‘ = 0 ( n # O ) . (5.15 b) 
Thus, as claimed, the n # 0 tensor modes h t ; are indeed massive, and turn out to have 
masses m, as in (5.1). Equations ( 5 . 1 3 ~ ) and ( 5 . 1 4 ~ ) show that the surviving scalar 
field h g ) and vector potential h:; are both massless. Also, by defining 
(5.16) (0) = h(0) +I 5 ( 0 ) i,”- ,U 2 7 7 e L Y h S 
(5.17) 
which is the equation of a massless spin-2 (graviton) field (Weinberg 1972). 
The masslessness of the n = 0 modes was anticipated in § § 1.4 and 1.7, where we 
made the ansatz of discarding the 6 dependence of all fields, as in equation (1.7) for 
example. In fact if we truncate the theory by retaining only the n = 0 modes, perform 
a Weyl rescaling (1.31), and carry out the 0 integration in (1.12), we obtain an effective 
four-dimensional action 
where K~ is defined in (1.16) and (1.17), 4“) is defined in (1.19), g:;, A:’ and hence 
F:: are defined analogously to as in (5.11)? and indices are raised and lowered 
with g:?. This (truncated) action is derived in Appelquist and Chodos (1983a, b), for 
example. The action (5.18) is consistent with the previous results (1.18) and (1.32). 
The masslessness of A, derives from the invariance of (5.18) under the gauge transfor- 
mation (1.11): 
A, + A: = A , +a,& (5.19) 
which in turn derives from the invariance of the original five-dimensional action under 
the (particular) coordinate transformation (1.9): 
e + e’= e+(&(x) ( 5 . 2 0 ~ ) 
xfi + X I @ = xp. (5.20b) 
In the same way the masslessness of gf: derives from the invariance of the action 
under the generalised (four-dimensional) coordinate transformation 
x F +D x”* = x, + 5” (x) (5.21 a ) 
e+ e t = e. (5.21b) 
1 1 1 8 D Bailin and A Love 
However (5.18) is also invariant under a global scale transformation, in which 
( 5 . 2 2 ~ ) ( 0 ) ! ( O ) = ( 0 ) g , v + g , v g P u 
A:) ~ A : O ) = A:) + AA:) (5.22b) 
4(0)+ + / ( O ) = 4 ( o ) - 2 A 4 ( 0 ) ( 5 . 2 2 ~ ) 
with A infinitesimal and constant. The ground state of the system has 
(gl"?) = 77PY (A:') = 0 (p) = E' (5.23) 
where r lWy is the Minkowski metric. Thus it has symmetry P 4 x R ' , where P4 is the 
four-dimensional Poincark group and RI is the gauge symmetry, which is not a compact 
U(1) because the truncated theory has no memory of the periodicity in 0. Since (r,b(O)) 
is non-zero, the ground state is not invariant under the global scale transformation. 
Thus the masslessness of 4") is because it is the Goldstone boson associated with the 
spontaneous breakdown of the global scale invariance (Dolan and Duff 1984). 
It is important to remember that the results for the spectrum of the five-dimensional 
Kaluza-Klein theory derived so far determine only the classical mass values. Quantum 
effects can modify these values. In particular, we should expect that the massless 
modes will only be 'truly' massless if their masslessness is protected by some sort of 
symmetry. In the case of the graviton field g:: and of the gauge field A:' this is indeed 
the case, since the gauge symmetries from which they derive are symmetries of the f i r 1 1 
theory. However the (Brans-Dicke) scalar field 4") is not so protected. The global 
scale invariance, of which it is the Goldstone boson, is a symmetry only of the truncated 
theory, in which the n # 0 modes are discarded. We shall see in the following section 
that the full theory does not have this symmetry. So 4") is only a pseudo-Goldstone 
boson (Dolan and Duff 1984). Its classical mass is zero, but we would expect that 
radiative corrections will shift the mass to a non-zero value. Evidently, an understand- 
ing of the symmetry of the full (untruncated) theory is necessary if we are to accurately 
determine the zero modes in a general Kaluza-Klein theory. However, before address- 
ing that task (in the next section) we shall describe briefly the work which has been 
carried out to determine the classical mass spectrum in more realistic, and therefore 
higher-dimensional, theories. 
The procedure is essentially a generalisation of that which we have described for 
the five-dimensional case. Firstly, one has to find a ground-state solution of the field 
equations, as in (5.6). Secondly, one considers arbitrary fluctuations of all fields about 
this solution, as in ( 5 . 7 ) , and expands these fluctuations, as in ( 5 . 1 1 ) , in a complete 
set of harmonics on the compact manifold. The coefficients of these harmonics are 
the physical (four-dimensional) fields, and substitution into the full higher-dimensional 
field equations yields the four-dimensional field equations, and hence the spectrum. 
However, there are a number of technical complications, which stem from the higher 
dimensionality, which deserve comment. 
Firstly, the ground-state metric (2.6) 
(O/gABIO) = diag(V,,, -im,,(.v)) (5.24) 
where 77," is the metric of flat Minkowski space M4, given in (1,4), and i,,(y) is the 
metric of the compact manifold, is in general not a solution of the (4+ D)-dimensional 
pure gravity field equations. This was discussed in 3. To achieve compactification 
it is necessary to introduce additional (matter) fields (or to allow a non-minimal 
gravitational action). The additional fields, of course, also have fluctuations about 
Kaluza- Klein theories 1119 
their ground-state (background field) values, and these fluctuations too must be expan- 
ded in terms of a complete set of harmonics on the compact manifold. Thus there are 
additional contributions to the spectrum, besides those originating in the purely 
gravitational sector. In general the final spectrum depends in an essential way upon 
the precise compactification mechanism actually utilised. 
The second complication occurs because the ‘harmonic expansion’ is in general 
considerably more complicated than in the five-dimensional case (5.11), and the 
quantum numbers of the mass eigenstates are complicated by the general non-Abelian 
nature of the isometry group of the compact manifold. The harmonic expansion (5.11) 
expands the fields in terms of the harmonics e ine which form a complete set of 
representations of the isometry group U(1) on the compact manifold S ’ . In general 
the fields of the theory transform as representations lR) of the tangent space group 
GT. For the case of a D-dimensional compact manifold K , the tangent space group is 
(5.25) 
The ‘harmonic expansion’ is then an expansion of the representations IR) in terms of 
the representations IG,) of the isometry group G I of the compact manifold K : 
GT = SO( 1 , 3 + D ) . 
(5.26) 
We shall assume that K is a coset space 
K = G / H (5.27) 
where G is a Lie group and H is a subgroup of G. In the case that H is a maximal 
subgroup of G, the isometry group is G itself; but if H is a non-maximal subgroup, 
then the isometry group is (Castellani er al 1984c) 
(5.28) 
where N‘(H) is the normaliser of H in G, but with any U( l ) factors common with G 
deleted. Thus in any event H is always a subgroup of G I so we may expand the 
representations IG,) of G, in terms of the representations Ih,) of H: 
G , = G x N’( H) 
(5.29) 
Also, we showed in (2.56) that H is a subgroup of the tangent space SO(D) of the 
compact manifoldK. Thus, since 
G T ~ S O ( ~ , ~ ) X S O ( D ) 
2 SO( 1,3) x H (5.30) 
we may also expand the representations lR) of G, in terms of the representations of H: 
(5.31) 
(5.32) 
This shows that the isometry group representations which actually occur in the harmonic 
expansion of lR) are those for which there is at least one overlap between the expansions 
(5.29) and (5.31) in representations of H (de Alwis and Koh 1984). The degeneracy 
of (G,) is given by the number of overlaps. 
1120 D Bailin and A Love 
As an illustration consider the ( D = 2) case in which 
K = S 2 = S0(3)/S0(2) SU(2)/U( 1). (5.33) 
Since H = SO(2) is the maximal subgroup of G = S0(3), the isometry group is the 
‘rotation’ group 
GI = SO(3). (5.34) 
Suppose also that we are concerned with the vector representation /vector,) of SO( 1,5). 
Under the decomposition (5.30) the expansion (5.3 1) becomes 
(5.35) ]vector,) = /vector,, 0) + IscaIar,, + 1) + /scalar,, - 1) 
using the notation ILorentz group representation, U( 1) charge). Consider first the 
scalar component with charge +l. Then the representations of the isometry group 
SO(3) which actually occur in the harmonic expansion are those which contain an 
element with U ( l ) charge t-1. Denoting the representations of SO(3) by their ‘angular 
momentum’ J, so that the dimensionality is 2 J + 1 , this means that the required 
representations are those with J integral and non-zero: 
J = n ( n = 1,2 ,3 , . . .). (5.36) 
The inclusion of the vector component with charge 0 extends this to include also the 
J = 0 representations of GI. 
The ‘realistic’ cases studied have for the most part been in the context of eleven- 
dimensional supergravity, in which the compactification is achieved using a background 
‘three-index photon’ field A B C D with field strength FABCD, defined in (3.10). As 
explained in 3 3.2, since supersymmetry forbids an eleven-dimensional cosmological 
constant, these supergravity theories typically have a classical ground state which is a 
four-dimensional anti-de Sitter space times a compact manifold K. The known solutions 
of the field equations fall into two classes: the Freund-Rubin (1978, 1980) solution 
and the Englert (1982) solution. In both classes the gravitino field vanishes; the 
four-dimensional Riemann tensor is maximally symmetric: 
R,,, = - k k , , ( x ) g , A x ) - g p A x ) g y p ( x ) l (5.37) 
where g F Y ( x ) is the anti-de Sitter space metric, and the ‘photon’ field strength in the 
4-space is given by 
(5.38) 
as in (3.14). Also, in both classes the metric and the field strength with mixed indices 
vanish: 
(5.39) 
Fpypu = ldet gl-1’2epLypu F 
g p m - - Ffivpm = Fwumn = FFmnp = 0. 
However, in the Freund-Rubin solution, 
Fmflpq(.Y) = 0 R m n = f F2imn k = f F 2 (5.40) 
whereas the Englert solution has 
F m n p q ( y ) = *$ FriTmnpqT R,, = 2 F2&,,, k = L F 2 12 (5.41) 
where 17 is a Killing spinor and T, , ,~~ is the totally antisymmetric product of four 
seven-dimensional Dirac matrices. All fields are then fluctuations about these ground- 
state values and these fluctuations may be decomposed into irreducible representations 
Kaluza- Klein theories 1121 
of the tangent space group SO(7) of the compact manifold K . For example, the 
fluctuations hAB(x, y ) of the metric may be decomposed into 1-, 7- and 27-dimensional 
representations, the last one being the symmetric traceless representation, and the 
fluctuations aABc(x, y ) in the 'photon' field may be decomposed into 1-, 7-, 21- and 
35-dimensional representations. We may then perform the harmonic expansion (5.26) 
for each of these tangent space group representations. This expresses the various 
fluctuations in terms of a basis of these irreducible representations of G satisfying the 
criteria described after (5.32). We call these basis elements Y"i(y) 'spherical' har- 
monics, where N I , N 7 , N I 7 , etc, identify representations of the tangent space group 
from which the harmonic arose; the harmonics are eigenfunctions of the H-invariant 
d' Alembertian operator 
AyYNc(y ) = - p N ~ Y N ~ ( y ) (5.42) 
where Ay is the Hodge-de Rham operator. The various field equations are linearised 
in the fluctuations, as in (5.8), and after fixing the gauge the mass eigenstates are 
identified. The mass eigenvalues are specified in terms of the eigenvalues p*" of the 
invariant operators on the internal space. The general solution in the case of the 
Freund-Rubin solution has been derived by Castellani et a1 (1984b), but in the case 
of the Englert solution a calculation of the bosonic spectrum is still awaited. (It is not 
known whether these are the only solutions.) To fix the numerical values of the mass 
eigenstates it is necessary to specify precisely which coset space G / H is being con- 
sidered. In the case of the round 7-sphere the general treatment reproduces the 
previously found results (Biran et a1 1983, 1984, Duff and Pope 1983). In addition to 
the zero modes expected in an N = 8 supergravity theory (namely, one massless 
graviton, 28 massless SO(7) gauge vector bosons, 35 scalars and 35 pseudoscalars) 
there are an additional 294 massless scalars. It is not known whether these scalars 
will remain massless when quantum effects are included, since their masslessness is 
not obviously protected by a gauge symmetry. The results for the squashed 7-sphere 
were derived by Awada et a1 (1983), Bais et a1 (1983) and by Nilsson and Pope (1984). 
The complete bosonic spectrum for the Mpqr solutions (Witten 1981) of eleven- 
dimensional supergravity, with Freund-Rubin compactification, has been found by 
D'Auria and FrC (1984a, b). 
5.2. Kac- Moody symmetries 
We saw in the previous section that the Brans-Dicke scalar field 4'') of the five- 
dimensional Kaluza-Klein theory was massless (only) at the classical level because of 
the scale invariance of the truncated theory. This masslessness is not expected to 
survive beyond the classical approximation, since the scale invariance is not an 
invariance of the complete theory. We now wish to analyse the symmetry of the theory 
in four dimensions when the complete tower of massive states is retained. This derives 
from the general five-dimensional coordinate invariance. Under a general 
(infinitesimal) coordinate transformation 
xw -, xw + y ( x , e) ( 5 . 4 3 ~ ) 
e + e+15(x , e ) (5.43 b ) 
where 
(5.43 c) 
1122 D Bailin and A Love 
(5.43 d ) 
As in (1.19), and (5.11), we are restricted to periodic variation of the coordinates, 
because of the topology of the ground state. Then it is easy to see that the global scale 
transformation (5.22) is not a symmetry of the complete theory, since to effect the 
transformation it is required to rescale the metric 
E A R + (112A/3)gAB (5.44) 
and combine this with the general coordinate transformation in which 
LA = 8$(-AdO) . (5.45) 
Clearly this coordinate transformation does not satisfy the periodicity requirement 
(5.43c), which means that 4") is only a pseudo-Goldstone boson, as claimed in the 
previous section. 
In ordinary four-dimensional relativity the PoincarC invariance may be regarded 
as a special case of the general covariance in which the infinitesimal coordinate 
transformations l ' " (x ) are restricted to the linear form 
(5.46) 
where U'" and up{, = -wVp are constant. To find the analogue in our five-dimensional 
theory we restrict the quantities l ' " ' A ( ~ ) in an analogous manner 
['*(x) = a'" + w'",xy 
~ ( " J F ( ~ ) = a ( n ) ~ + w ( " ) ~ "X (5.47a ) 
p 5 ( X ) = C(" ) (5.47 b) 
where a'" ' , w""" and c(') are constants. We may determine the generators associated 
with these constants in the usual way. Consider, for example, a translation with all 
w ( ~ ' , c(" ) zero and a single non-zero a'"'",